Waveform Design For The Integrated Sensing, Communication and Simultaneous Wireless Information And Power Transfer System

1. Introduction

2. Materials and Methods

2.1. The CP-OTFS Model

2.1.1. Basic Concepts of CP-OTFS

The time–frequency (TF) plane is discretized to a $N\times M$ grid as following:

$$\Delta =(nT,m\Delta f),n=0,...,N-1,m=0,...,M-1$$

(1)

where T and $\Delta f=1/T$ are sampling intervals of time and frequency axes, respectively.

The modulated TF samples $\chi [n,m]$ are transmitted over an OTFS frame with duration $\widehat{T}=NT$ and occupy a bandwidth $B=M\Delta f$. The sampling frequency $f_s=B$.

The DD plane is discretized to an $N\times M$ lattice as following:

$$\Xi =(\frac{k}{NT},\frac{l}{M\Delta f}),k=0,...,N-1,l=0,...,M-1.$$

(2)

2.1.2. Input-Output Relationship of CP-OTFS Based ISWSCPTS

   A. The transmitter model

The CP-OTFS based system is depicted in Figure 2. The symbols $\chi [k,l]$ in DD domain is first transformed into symbols $\widehat{\chi }[n,m]$ in TF domain by the inverse symplectic finite Fourier transform (ISFFT). Then, the Heisenberg transform is applied to $\widehat{\chi }$ to generate time domain discrete signal $\widehat{x}\left[i\right],i=0,...,NM-1$. The expression of $\widehat{x}\left[i\right]$ is

$$\begin{array}{cc}\hfill \widehat{x}\left[i\right]=& \hfill \sum _{n^{\prime }}^{N-1}\sum _{m^{\prime }}^{M-1}\sum _{k^{\prime }}^{N-1}\sum _{l^{\prime }}^{M-1}\chi [k,l]\hfill \\ & \hfill \exp \left\{j2\pi (\frac{k^{\prime }{n}^{\prime }}{N}-\frac{l^{\prime }{m}^{\prime }}{M}+\frac{m^{\prime }i}{M})\right\}{g}_{tx}[i-n^{\prime }M],\hfill \end{array}$$

(3)

where the $g_{tx}\left[\iota \right]$ is the transmitting rectangular pulse and its expression is

$$g_{tx}\left[\iota \right]=\left\{\begin{array}{c}\hfill 1,0\le \iota \le M-1\hfill \\ \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(4)

Next, the discrete CP-OTFS transmitting signal $x\left[i\right]$ is constructed by adding a CP of length M to $x\left[i\right]$. Finally, $x\left[i\right]$ is converted to radio frequency (RF) signal $x\left(t\right)$ by RF transmitter module.

   B. The channel model

In the application scenarios of ISWSCPTS, the perception target, communication channel, and energy transmission channel can all be modeled using a unified framework based on the DD channel (DDC) model. Assuming there are P DDCs which can be expressed as:

$$\rho (\tau ,f_d)=\sum _{p=1}^P{\alpha }_p\delta (\tau -{\tau }_p)\delta (f_d-f_{d_p}),$$

(5)

where ${\alpha }_p$ is the channel gain coefficient, $f_{d_p}$ is the Doppler, and ${\tau }_p$ is the delay. Let $\alpha ={[{\alpha }_1,...,{\alpha }_p]}^{\mathrm{T}}$.

In this work, we make the following assumptions:

$$\begin{array}{cc}\hfill & \hfill {\tau }_p=l_p/M\Delta f,l_p=0,1,...,M-1\hfill \\ \hfill & \hfill f_{d_p}={\left(k_p\right)}_N/NT,k_p=0,1,...,N-1\hfill \end{array}$$

(6)

where $l_p$ and $k_p$ are integers.

Thus, the discrete form of $\rho (\tau ,f_d)$ can be expressed as:

$$\rho [k,l]=\sum _{p=1}^P{\alpha }_p\delta [k-{\left(k_p\right)}_N]\delta [l-l_p]$$

(7)

where

$${\left(k_p\right)}_N=\left\{\begin{array}{c}\hfill k_p,k_p\le N/2\hfill \\ \\ \hfill N-k_p,otherwise\hfill \end{array}\right.$$

(8)

   C. The receiver model

The received signal $r\left(t\right)$ is transformed into the discrete-time signal $r\left[i\right],i=0,...,NM-1,$ by RF receiver module and removing the CP. $r\left[i\right]$ can be expressed as:

$$r\left[i\right]=\sum _{p=1}^P{\alpha }_{p}x\left[{[i-l_p]}_{NM}\right]\exp \left\{j2\pi f_{d_p}(\frac{iT}{M}+T)\right\}+\widehat{w}\left[i\right]$$

(9)

where ${[•]}_{NM}$ denotes modulo $NM$ operation, $\widehat{w}\left[i\right]$ is the additive Gaussian white noise (AWGN).

$r\left[i\right]$ is converted to received symbols $\widehat{{\rm Y}}[n,m]$ in TF domain through the Wigner transform. $\widehat{{\rm Y}}[n,m]$ can be expressed as:

$$\widehat{{\rm Y}}[n,m]=\sum _{i=0}^{NM-1}r\left[i\right]g_{rx}^{*}[i-nM]\exp \left(j2\pi \frac{mi}{M}\right)$$

(10)

where $g_{rx}\left[\iota \right]$ is the receiving rectangular pulse and its expression is

$$g_{rx}\left[\iota \right]=\left\{\begin{array}{c}\hfill 1,0\le \iota \le M-1\hfill \\ \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(11)

Finally, $\widehat{{\rm Y}}[n,m]$ is converted into the received symbols ${\rm Y}[k,l]$ in DD domain through the symplectic finite Fourier transform (SFFT). ${\rm Y}[k,l]$ can be expressed as:

$${\rm Y}[k,l]=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}\widehat{{\rm Y}}[n,m]\exp \{-j2\pi (\frac{kn}{N}-\frac{lm}{M})\}+\mathbf{W}[k,l]$$

(12)

where $\mathbf{W}$ is AWGN matrix.

After laborious derivations, (12) can be recast as:

$${\rm Y}[k,l]=\sum _{k^{\prime }=0}^{N-1}\sum _{l^{\prime }=0}^{M-1}\chi [k^{\prime },l^{\prime }]H_{k,l}[k^{\prime },l^{\prime }]+\mathbf{W}[k,l]$$

(13)

where the expression of $H_{k,l}[k^{\prime },l^{\prime }]$ is as follows:

$$H_{k,l}[k^{\prime },l^{\prime }]=\frac{1}{N{M}^2}(H_{ISI}+H_{ICI})\exp \left(j2\pi f_{d}T\right)$$

(14)

In (14), $H_{ISI}$ and $H_{ICI}$ are expressed as follows:

(15)

(16)

According to (13), each receiving symbol is the result of summing all input symbols weighted by $H_{k,l}[k^{\prime },l^{\prime }]$. Clearly, $H_{k,l}[k^{\prime },l^{\prime }]$ is related to variables l, l, $k^{\prime }$, and $l^{\prime }$. The magnitude response of $H_{k,l}[k^{\prime },l^{\prime }]$ is as follows:

$$|H_{k,l}[k^{\prime },l^{\prime }]|=\left\{\begin{array}{c}\hfill 1,k^{\prime }={[k-k_p]}_N,l^{\prime }={[l-l_p]}_M\hfill \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(17)

Therefore, the summation operation in (13) can be removed, and it can be simplified as follows:

$${\rm Y}[k,l]=\sum _{p=1}^P{\alpha }_p\chi [{[k-k_p]}_N,{[l-l_p]}_M]H_{k,l}[k_p,l_p]+\mathbf{W}[k,l],$$

(18)

where $H_{k,l}[k_p,l_p]$ is defined in (21), the definitions of $H_1$ and $H_2$ are (19) and (20), respectively. Note that (19)–(21) are exact expressions without any approximation.

$$\begin{array}{c}\hfill H_1=N\sum _{m=0}^{M-1}\sum _{m^{\prime }=0}^{M-1}\exp \{-j2\pi \frac{l(m^{\prime }-m)}{M}\}\sum _{i^{\prime }=l_p}^{M-1}\exp \left\{j2\pi \frac{m^{\prime }-m+f_{d_p}T}{M}\right\}\end{array}$$

(19)

$$\begin{array}{c}\hfill H_2=N\exp (-j2\pi \frac{{[k-k_p]}_N}{N})\sum _{m=0}^{M-1}\sum _{m^{\prime }=0}^{M-1}\exp \{-j2\pi \frac{l(m^{\prime }-m)}{M}\}\sum _{i^{\prime }=0}^{l_p-1}\exp \left\{j2\pi \frac{m^{\prime }-m+f_{d_p}T}{M}\right\}\end{array}$$

(20)

According to (3)–(12), after tedious derivations, ${\rm Y}[k,l]$ can be expressed as:

$$H_{k,l}[k_p,l_p]=\frac{1}{N{M}^2}(H_1+H_2)\exp \left(j2\pi f_{d_p}T\right)$$

(21)

3. Results and Discussions

4. Conclusions

Abbreviations

References

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